For \(t\in \mathbb {T}\) , \(\mathbf {v}^+=(\mathbf {i}^+,\mathbf {j}^+)\in \Sigma _{a,b}^+\)  and \(n\ge 0\)  define the following rectangle centred at \(\Phi (\mathbf {v}^+)\in \mathbb {R}^2\) ,
\(U_{n,t}(\mathbf {v}^+)=B(\Phi _{\mathbf {I}}(\mathbf {i}^+),R_{\mathbf {I}}\delta ^{\tau _n(t)})\times B(\Phi _{\mathbf {J}}(\mathbf {j}^+),R_{\mathbf {J}}\rho ^{n}),\) 
where we set \(\tau _0(t)=0\) . By the fact that \(|\tau _n(t)-n\alpha |\le 2\)  we have for all \(t\in \mathbb {T}\) , \(\mathbf {v}^+=(\mathbf {i}^+,\mathbf {j}^+)\in \Sigma _{a,b}^+\)  and \(n\ge 0\) ,
\(B(\Phi (\mathbf {v}^+),r\cdot \rho ^n )\subset U_{n,t}(\mathbf {v}^+)\subset B(\Phi (\mathbf {v}^+),R\cdot \rho ^n),\) 
where \(r=\delta ^2\cdot \min \lbrace R_{\mathbf {I}},R_{\mathbf {J}}\rbrace \)  and \(R=\delta ^{-2}\max \lbrace R_{\mathbf {I}},R_{\mathbf {J}}\rbrace \) . Denote by
\(\gamma =\frac{R}{r}=\delta ^{-4}\frac{\max \lbrace R_{\mathbf {I}},R_{\mathbf {J}}\rbrace }{\min \lbrace R_{\mathbf {I}},R_{\mathbf {J}}\rbrace }\ge 1.\) 
Then it is easy to see that the family
\(\mathcal {U}=\lbrace U_{n,t}(\mathbf {v}^+): t\in \mathbb {T},\ \mathbf {v}^+=(\mathbf {i}^+,\mathbf {j}^+)\in \Sigma _{a,b}^+,\ n\ge 0\rbrace \) 
is a fine \(\gamma \) -Morse covering of \(E=\Phi (\Sigma _{a,b}^+)=\Phi _{\mathbf {I}}(\Sigma _a^+)\times \Phi _{\mathbf {J}}(\Sigma _b^+)\) . For \(U_{n,t}(\mathbf {v}^+)\in \mathcal {U}\)  shortly denote by
\(U^\Phi _{n,t}(\mathbf {v}^+)=\Phi ^{-1}(U_{n,t}(\mathbf {v}^+))\) 
the corresponding set in \(\Sigma _{a,b}^+\) .
For \(u\in \Sigma _{a,b}^{+,*}\)  and \(B\in \mathcal {B}_{a,b}\)  denote by
\(uB=\lbrace u\mathbf {v}^+:\mathbf {v}^+\in B\rbrace .\) 
Recall the notation \(C_{n,t}\)  in (REF ). For \(t\in \mathbb {T}\)  define
\(\mathcal {P}_t=\lbrace [u]:u\in C_{1,t}\rbrace .\) 
Applying [1]} to \((\Sigma _a^+, \Phi _{\mathbf {I}})\)  and \((\Sigma _b^+, \Phi _{\mathbf {J}})\)  respectively we obtain that for \(n\ge 1\) ,
\(U^\Phi _{n,t}(\mathbf {v}^+)\cap \mathcal {P}_t(\mathbf {v}^+)=\mathbf {v}^+|_{1,t}U^\Phi _{n-1,R(t)}(\sigma _t\mathbf {v}^+).\) 
By (REF ) we have
\(\widetilde{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}(\mathbf {v}^+|_{k,t}B)=Q_{\mathbf {v}^+|_{k,t}}(\mathbf {\omega })\cdot \theta _{t,\mathbf {v}^-}^+([\mathbf {v}^+|_{k,t}])\cdot \widetilde{\theta }_{R^k(t),\mathbf {v}^-\mathbf {v}^+|_{k,t},\mathbf {\omega }_{\mathbf {v}^+|_{k,t}}}^{+}(B)\) 

Therefore
\(&\ \frac{\widetilde{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}(\mathbf {v}^+|_{k+1,t}U^\Phi _{n-k-1,R^{k+1}(t)}(\sigma _{k+1,t}\mathbf {v}^+))}{\widetilde{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}(\mathbf {v}^+|_{k,t}U^\Phi _{n-k,R^k(t)}(\sigma _{k,t}\mathbf {v}^+))}\\=&\ \frac{\widetilde{\theta }^+_{R^k(t),\mathbf {v}^-\mathbf {v}^+|_{k,t},\mathbf {\omega }_{\mathbf {v}^+|_{k,t}}}(\sigma _{k,t}\mathbf {v}^+|_{1,R^{k}t}U^\Phi _{n-k-1,R^{k+1}(t)}(\sigma _{k,t}\mathbf {v}^+))}{\widetilde{\theta }^+_{R^k(t),\mathbf {v}^-\mathbf {v}^+|_{k,t},\mathbf {\omega }_{\mathbf {v}^+|_{k,t}}}(U^\Phi _{n-k,R^k(t)}(\sigma _{k,t}\mathbf {v}^+))}\\=&\ \frac{\bar{\theta }^+_{R^k(t),\mathbf {v}^-\mathbf {v}^+|_{k,t},\mathbf {\omega }_{\mathbf {v}^+|_{k,t}}}(U^\Phi _{n-k,R^k(t)}(\sigma _{k,t}\mathbf {v}^+)\cap \mathcal {P}_{R^k(t)}(\sigma _{k,t}\mathbf {v}^+))}{\bar{\theta }^+_{R^k(t),\mathbf {v}^-\mathbf {v}^+|_{k,t},\mathbf {\omega }_{\mathbf {v}^+|_{k,t}}}(U^\Phi _{n-k,R^k(t)}(\sigma _{k,t}\mathbf {v}^+))}\\\) 

Define
\(F_n(t,\mathbf {v}^-,\mathbf {v}^+,\mathbf {\omega })=-\log \frac{\bar{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}(U^\Phi _{n,t}(\mathbf {v}^+)\cap \mathcal {P}_t(\mathbf {v}^+))}{\bar{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}(U^\Phi _{n,t}(\mathbf {v}^+))}\) 
Then we have
\(\log \frac{\widetilde{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}(U^\Phi _{n,t}(\mathbf {v}^+))}{\widetilde{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}([\mathbf {v}^+|_{n,t}])}=\sum _{k=0}^{n-1}F_{n-k}\circ \widetilde{T}^k(t,\mathbf {v}^-,\mathbf {v}^+,\mathbf {\omega }),\) 
where we have used the fact that, by the choices of \(R_{\mathbf {I}}\) , \(R_{\mathbf {J}}\) , \(U^\Phi _{0,t}(\mathbf {v}^+)=\Sigma _{a,b}^+\)  is the whole set for all \(t\in \mathbb {T}\)  and \(\mathbf {v}^+\in \Sigma _{a,b}^+\) .
By Proposition REF , for \(\mathbb {Q}_\theta ^-\) -a.e. \((t,\mathbf {v}^-,\mathbf {\omega })\) , for \(\bar{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}\) -a.e. \(\mathbf {v}^+\) , as \(n\rightarrow \infty \) , the sequence \(F_n(t,\mathbf {v}^-,\mathbf {v}^+,\mathbf {\omega })\)  converges to
\(F(t,\mathbf {v}^-,\mathbf {v}^+,\mathbf {\omega })=I_{\bar{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}}(\mathcal {P}_t|\mathcal {B}_{\mathbf {I}}\otimes \mathcal {B}_{\mathbf {J}})(\mathbf {v}^+)\) 
and there exists a positive measurable function \(g(t,\mathbf {v}^-,\cdot ,\mathbf {\omega })\in L^1(\Sigma _{a,b}^+,\mathcal {B}_{a,b}^+,\bar{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }})\)  such that \(F_n\le g\)  and
\(\int _{\Sigma _{a,b}^+} g(t,\mathbf {v}^-,\mathbf {v}^+,\mathbf {\omega })\,\bar{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}(\mathrm {d}\mathbf {v}^+) \le ab\log ab+C<\infty .\) 
By using the exact-dimensionality of \(\widetilde{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}\)  in (REF ) and using Theorem REF  we deduce that for \(\mathbb {Q}_\theta \) -a.e. \((t,\mathbf {v},\mathbf {\omega })\) , or equivalently (by (REF )), for \(\mathbb {Q}_\theta ^-\) -a.e. \((t,\mathbf {v}^-,\mathbf {\omega })\) , for \(\bar{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}\) -a.e. \(\mathbf {v}^+\) ,
\(\lim _{n\rightarrow \infty }\frac{-\log \widetilde{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}(U^\Phi _{n,t}(\mathbf {v}^+))}{n}&=\ \lim _{n\rightarrow \infty }\frac{-\log \widetilde{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}([\mathbf {v}^+|_{n,t}]) }{n}-\frac{1}{n}\sum _{k=0}^{n-1} F_{n-k}\circ \widetilde{T}^k(t,\mathbf {v}^-,\mathbf {v}^+,\mathbf {\omega })\\&=\ \alpha (h_\mu -h_V)+(h_\nu -h_W)-\mathbb {E}_{\mathbb {Q}_\theta }(F).\) 
Recall that \(\theta =\Theta ^\zeta _z\)  is the ergodic decomposition of \(\Theta \)  at a \(\Theta \) -typical \(z\in X\) . By Lemma REF  we may write
\(\mathbb {E}_{\mathbb {Q}_\Theta }(f)=\int _{\mathcal {M}(X)}\mathbb {E}_{\mathbb {Q}_{\theta }}(f)\,\Theta (\mathrm {d}\theta )\) 
for any measurable function \(f\)  on \((\widetilde{X},\widetilde{\mathcal {F}})\) . Hence by (REF ) and (REF ) we have that
\(\begin{split}\int _{\mathbb {T}\times \Sigma _{a,b}^-\times \Omega _{a,b}}\mathbb {E}_{\bar{\lambda }^+_{\mathbf {v}^-,\mathbf {\omega }}}(f^+)&\,\mathbb {Q}_\Theta ^-(\mathrm {d}(t,\mathbf {v}^-,\mathbf {\omega }))\\=&\ \int _{\mathcal {M}(X)}\int _{\mathbb {T}\times \Sigma _{a,b}^-\times \Omega _{a,b}}\mathbb {E}_{\bar{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}}(f^+)\,\mathbb {Q}_{\theta }^-(\mathrm {d}(t,\mathbf {v}^-,\mathbf {\omega })) \Theta (\mathrm {d}\theta )\end{split}\) 
for any measurable function \(f^+\)  on \((\Sigma _{a,b}^+,\mathcal {B}_{a,b}^+)\) . By the definition of lower Hausdorff dimension we deduce that the dimension of \(\bar{\lambda }^+_{\mathbf {v}^-,\mathbf {\omega }}\)  on \((\Sigma _{a,b}^+,d_{\delta ,\rho }^+)\)  can not be lower than the essential lower dimension of \(\bar{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}\) , which is
\(\frac{\alpha (h_\mu -h_V)+(h_\nu -h_W)-\mathbb {E}_{\mathbb {Q}_\theta }(F)}{-\log \rho }\) 
by (REF ). Comparing this with (REF ) we deduce that
\(\mathbb {E}_{\mathbb {Q}_\theta }(F)\ge \alpha h_{\mu ,V,\mathbf {I}} + h_{\nu ,W,\mathbf {J}}.\) 
We have that
\(\mathbb {E}_{\mathbb {Q}_\theta }(F)=\int _{\Sigma _{a,b}^-\times \Omega _{a,b}} H_{\bar{\theta }_{t,\mathbf {v}^-,\mathbf {\omega }}}(\mathcal {P}_t|\mathcal {B}_{\mathbf {I}}\otimes \mathcal {B}_{\mathbf {J}}) \,\mathbb {Q}_\theta ^-(\mathrm {d}(t,\mathbf {v}^-,\mathbf {\omega }))\) 
Recall that
\(h_{\mu ,V,\mathbf {I}}=\int _{\Sigma _a^-\times \Omega _a} H_{\bar{\mu }_{\mathbf {i}^-,\mathbf {\omega }^a}}(\mathcal {P}_a|\mathcal {B}_{\mathbf {I}}) \,\mathbb {Q}_\mu ^-(\mathrm {d}(\mathbf {i}^-,\mathbf {\omega }^a))\) 
\(h_{\nu ,W,\mathbf {J}}=\int _{\Sigma _b^-\times \Omega _b} H_{\bar{\nu }_{\mathbf {j}^-,\mathbf {\omega }^b}}(\mathcal {P}_b|\mathcal {B}_{\mathbf {J}}) \,\mathbb {Q}_\nu ^-(\mathrm {d}(\mathbf {j}^-,\mathbf {\omega }^b))\) 
Note that \(\mathcal {P}_t=\mathcal {P}_a\times \mathcal {P}_b\)  if \(t\in A\)  and \(\mathcal {P}_t=\lbrace \emptyset ,\Omega _a\rbrace \times \mathcal {P}_b\)  if \(t\in A^c\) . Therefore
\(&\ \alpha h_{\mu ,V,\mathbf {I}} + h_{\nu ,W,\mathbf {J}}\\=&\ \int _{\Sigma _{a,b}^-\times \Omega _{a,b}} H_{\bar{\mu }_{\mathbf {i}^-,\mathbf {\omega }^a}\times \bar{\nu }_{\mathbf {j}^-,\mathbf {\omega }^b}}(\mathcal {P}_t|\mathcal {B}_{\mathbf {I}}\otimes \mathcal {B}_{\mathbf {J}})\, \ell (\mathrm {d}t) \mathbb {Q}_\mu ^-(\mathrm {d}(\mathbf {i}^-,\mathbf {\omega }^a))\mathbb {Q}_\nu ^-(\mathrm {d}(\mathbf {j}^-,\mathbf {\omega }^b))\\=&\ \int _{\Sigma _{a,b}^-\times \Omega _{a,b}} H_{\bar{\lambda }_{\mathbf {v}^-,\mathbf {\omega }}}(\mathcal {P}_t|\mathcal {B}_{\mathbf {I}}\otimes \mathcal {B}_{\mathbf {J}})\, \mathbb {Q}_\Theta ^-(\mathrm {d}(t,\mathbf {v}^-,\mathbf {\omega }))\) 

Following the same approach as in the proof of [2]}, by the concavity of conditional entropy as a function of probability measures, and then by Jensen's inequality and (REF ) we deduce that
\(\alpha h_{\mu ,V,\mathbf {I}} + h_{\nu ,W,\mathbf {J}}\ge &\ \int _{\mathcal {M}(X)}\int _{\Sigma _{a,b}^-\times \Omega _{a,b}} H_{\bar{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }}}(\mathcal {P}_t|\mathcal {B}_{\mathbf {I}}\otimes \mathcal {B}_{\mathbf {J}})\, \mathbb {Q}_{\theta }^-(\mathrm {d}(t,\mathbf {v}^-,\mathbf {\omega })) \Theta (\mathrm {d}\theta )\\=&\ \int _{\mathcal {M}(X)} \mathbb {E}_{\mathbb {Q}_{\theta }}(F)\, \Theta (\mathrm {d}\theta )\) 

Together with (REF ) we deduce that for \(\Theta \) -typical \(\theta \) ,
\(\mathbb {E}_{\mathbb {Q}_{\theta }}(F)=\alpha h_{\mu ,V,\mathbf {I}} + h_{\nu ,W,\mathbf {J}}.\) 
Then we get the conclusion by (REF ).
